An introductory course that explores the fundamental concepts and techniques of mathematical analysis.
A series is what you get when you add up the terms of a sequence: \( a_1 + a_2 + a_3 + ... \).
A series converges if the sum approaches a real number as you add more and more terms. If not, it diverges.
There are cool tests to check if a series converges, like the comparison test or the ratio test.
Series are used to represent functions, calculate decimals of numbers like \( \pi \), and solve problems in physics and engineering.
The geometric series \( 1 + \frac{1}{2} + \frac{1}{4} + ... \) converges to 2!
The harmonic series \( 1 + \frac{1}{2} + \frac{1}{3} + ... \) diverges.
The alternating series \( 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + ... \) converges.
A series is a sum of infinitely many numbers, and convergence means the sum settles to a specific value.